Solvability Complexity Index Classification For Koopman Operator Spectra In $L^p$ For $1<p<\infty$

Christopher Sorg

Paper

Solvability Complexity Index Classification For Koopman Operator Spectra In $L^p$ For $1<p<\infty$

Abstract

We study the computation of the approximate point spectrum and the approximate point

-pseudospectrum of bounded Koopman operators acting on

for

and a compact metric space

with finite Borel measure

. Building on finite sections in a computable unconditional Schauder basis of

, we design residual tests that use only finitely many evaluations of the underlying map and produce compact sets on a planar grid, that converge in the Hausdorff metric to the target spectral sets, without spectral pollution. From these constructions we obtain a complete classification, in the sense of the Solvability Complexity Index. Also we analyze the sufficiency and existence of a Wold-von Neumann decomposition analog, that was used in the special

-case. The main difficulty in extending from the already analyzed Hilbert setting

to general

is the loss of orthogonality and Hilbertian structure: there is no orthonormal basis with orthogonal coordinate projections in general, the canonical truncations

in a computable Schauder dictionary need not be contractive (and may oscillate) and the Wold-von Neumann reduction has no directly computable analog in

. We overcome these obstacles by working with computable unconditional dictionaries adapted to dyadic/Lipschitz filtrations and proving stability of residual tests under non-orthogonal truncations.